Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

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Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

David Nualart, Bruno Saussereau

To cite this version:

David Nualart, Bruno Saussereau. Malliavin calculus for stochastic differential equations driven by

a fractional Brownian motion. Stochastic Processes and their Applications, Elsevier, 2009, 119 (2),

pp.391-409. �10.1016/j.spa.2008.02.016�. �hal-00485648�

Malliavin calculus for stochastic differential equations driven by a fractional Brownian

motion.

David Nualart

^∗

Department of Mathematics

University of Kansas Lawrence, Kansas, 66045, USA

Bruno Saussereau

D´ epartement de Math´ ematiques Universit´ e de Franche-Comt´ e

16, Route de Gray 25030 Besan¸con, France October 26, 2006

1 Introduction

Let B = {B

t

, t ≥ 0} be an m−dimensional fractional Brownian motion (fBm in short) of Hurst parameter H ∈ (0, 1). That is, B is a centered Gaussian process with the covariance function E (B

_sⁱ

B

_t^j

) = R

_H

(s, t)δ

_ij

, where

R

H

(s, t) = 1

2 t

^2H

+ s

^2H

− |t − s|

^2H

. (1)

If H =

¹₂

, B is a Brownian motion. From (1), it follows that E |B

t

− B

s

|

²

= m|t − s|

^H

so the process B has α-H¨ older continuous paths for all α ∈ (0, H).

We refer to [11] and references therein for further information about fBm and stochastic integration with respect to this process.

In this article we fix

¹₂

< H < 1 and we consider the solution {X

t

, t ∈ [0,T ]}

of the following stochastic differential equation on R

^d

X

_tⁱ

= x

ⁱ₀

+

m

X

j=1

Z

t 0

σ

^ij

(X

_s

)dB

_s^j

+ Z

t

0

b

ⁱ

(X

_s

)ds , t ∈ [0,T ] , (2)

i = 1, . . . , d, where x

0

∈ R

^d

is the initial value of the process X .

∗Partially supported by the NSF Grant DMS 0604207

The stochastic integral in (2) is a path-wise Riemann-Stieltjes integral (see Young [15]). Suppose that σ has bounded partial derivatives which are H¨ older continuous of order λ >

_H¹

−1, and b is Lipschitz, then there is a unique solution to Equation (2) which has H¨ older continuous trajectories of order H − ε, for any ε > 0. This result has been proved by Lyons in [6] in the case b = 0, using the p-variation norm. The theory of rough paths analysis introduced by Lyons in [7] was used by Coutin and Qian to prove an existence and uniqueness result for Equation (2) in the case H ∈ (

¹₄

,

¹₂

) (see [2]).

The Riemann-Stieltjes integral appearing in Equation (2) can be expressed as a Lebesgue integral using a fractional integration by parts formula (see Z¨ ahle [16]). Using this formula for the Riemann-Stieltjes integral, Nualart and R˘ a¸scanu have established in [12] the existence of a unique solution for a class of general differential equations that includes (2).

The main purpose of our work is to study the regularity of the solution to Equation (2) in the sense of Malliavin calculus, and to show the absolute continuity for the law of X

t

for t > 0, assuming an ellipticitity condition on the coefficient σ. First we establish a general result on the regularity with respect to the driven function for the solution of deterministic equations, using the techniques of fractional calculus developed in [12]. This allows us to deduce the differentiability of the solution to Equation (2) in the direction of the Cameron- Martin space. These results are related to those proved by Lyons and Dong Li in [8] on the smoothness of Itˆ o maps for such equations in term of Fr´ echet-Gˆ ateaux differentiability.

The regularity results obtained here have been used in a recent paper by Bau- doin and Hairer [1] to show the smoothness of the density under a hypoellipticity H¨ ormander’s condition. This result requires also the existence of moments for the iterated derivatives, which has been established in [4]. In [9], the existence of a density for the solution of a one-dimensional equation is shown.

The paper is organized as follows. In Section 2 we establish the Fr´ echet differentiability with respect to the input function for deterministic differential equations driven by H¨ older continuous functons. Section 3 is devoted to analyze stochastic differential equations driven by a fBm with Hurst parameter H ∈ (

¹₂

, 1), the main result being the differentiability of the solution in the directions of the Cameron-Martin space. In Section 4 we prove the absolute continuity of the solution under ellipticity assumptions. The proofs of some technical results are given in the Appendix.

2 Deterministic differential equations driven by rough functions

We first introduce some preliminaries. Given a measurable function f : [0,T ] → R

^d

and α ∈ (0,

¹₂

), we will make use of the notation

∆

^α_t

(f ) = |f (t)| + Z

t

0

|f (t) − f (s)|

|t − s|

^α+1

ds.

We denote by W

₁^α

(0, T ; R

^d

) the space of measurable functions f : [0,T ] → R

^d

such that

kf k

α,1

:= sup

t∈[0,T]

∆

^α_t

(f ) < ∞ .

For any 0 < λ ≤ 1, denote by C

^λ

(0, T ; R

^d

) the space of λ-H¨ older continuous functions f : [0,T ] → R

^d

, equipped with the norm

kf k

λ

:= kf k

_∞

+ sup

0≤s<t≤T

|f (t) − f (s)|

(t − s)

^λ

,

where kf k

∞

:= sup

_t∈[0,T]

|f (t)|. We denote by W

₂^1−α

(0, T ; R

^m

) the space of measurable functions g : [0,T ] → R

^m

such that

kgk

_1−α,2

:= sup

0≤s<t≤T

|g(t) − g(s)|

(t − s)

^1−α

+ Z

t

s

|g(y) − g(s)|

(y − s)

^2−α

dy

< ∞ . Clearly for any ε > 0 such that 1 − α+ ε ≤ 1 we have

C

^α+ε

(0,T ; R

^d

) ⊂ W

₁^α

(0, T ; R

^d

) and

C

^1−α+ε

(0, T ; R

^m

) ⊂ W

₂^1−α

(0, T ; R

^m

) ⊂ C

^1−α

(0, T ; R

^m

) . For d = m = 1 we simply write W

₁^α

(0, T ), C

^λ

(0,T ), and W

₂^1−α

(0, T ).

Suppose that g ∈ W

₂^1−α

(0, T ) and f ∈ W

₁^α

(0, T ). In [16], Z¨ ahle introduced the generalized Stieltjes integral

Z

T 0

f

_t

dg

_t

= (−1)

^α

Z

T

0

D

^α₀₊

f

(t) D

^1−α_T−

g

_T−

(t)dt, (3)

defined in terms of the fractional derivative operators D

^α₀₊

f(x) = 1

Γ(1 − α) f (x)

x

^α

+ α Z

x

0

f (x) − f (y) (x − y)

^α+1

dy

, and

D

_T^α₋

g

_T−

(x) = (−1)

^α

Γ(1 − α)

g(x) − g(T ) (T − x)

^α

+ α

Z

T x

g(x) − g(y) (x − y)

^α+1

dy

! . We refer to [13] for further details on fractional operators. Z¨ ahle proved that if f ∈ C

^α+ε

(0, T ), then this integral coincides with the Riemann-Stieltjes integral, which exists by the results of Young (see [15]). Using formula (3), Nualart and R˘ a¸scanu have derived the following estimates (see [12], Propositions 4.1 and 4.3).

Proposition 1 Fix 0 < α <

¹₂

. Given two functions g ∈ W

₂^1−α

(0, T ) and f ∈ W

₁^α

(0, T ), we denote G

t

(f ) = R

t

0

f

s

dg

s

and F

t

(f ) = R

t 0

f

s

ds.

(i) The function G(f ) belongs to C

^1−α

(0, T ) and we have

∆

^α_t

(G(f )) ≤ c

_α,T

kgk

_1−α,2

Z

t

0

(t − r)

^−2α

+ t

^−α

∆

^α_r

(f )dr , (4)

kG(f )k

_1−α

≤ c

_α,T

kgk

1−α,2

kf k

_α,1

, (5)

with a constant c

_α,T

which depends only on α and T . (ii) The function F (f ) belongs to C

¹

(0, T ) and moreover

∆

^α_t

(F (f )) ≤ c

α,T

Z

t 0

|f

s

|

(t − s)

^α

ds , (6)

kF (f )k

₁

≤ c

_T

kf k

_∞

, (7) with a constant c

_α,T

which depends only on α and T .

We first study deterministic differential equations driven by H¨ older con- tinuous functions of order stricktly larger that

¹₂

. Fix 0 < α <

¹₂

. Let g ∈ W

₂^1−α

(0, T ; R

^m

) and consider the deterministic differential equation on R

^d

x

ⁱ_t

= x

ⁱ₀

+ Z

t

0

b

ⁱ

(x

s

)ds +

m

X

j=1

Z

t 0

σ

^ij

(x

s

)dg

_s^j

, t ∈ [0,T ] , (8) i = 1, . . . , d, where x

0

∈ R

^d

.

For any integer k ≥ 1 we denote by C

_b^k

the class of real-valued functions on R

^d

which are k times continuously differentiable with bounded partial deriva- tives up to the kth order. We also denote by C

_b^∞

the class of infinitely differen- tiable functions on R

^d

with bounded partial derivatives of all orders.

In [12], the authors prove that Equation (8) has a unique solution x ∈ W

₁^α

(0, T ; R

^d

) which is moreover (1 − α)-H¨ older continuous, if b

ⁱ

, σ

^ij

∈ C

_b¹

and the partial derivatives of σ

^ij

are H¨ older contiuous of order λ >

_H¹

− 1.

In this section we will show the differentiability of the mapping g → x(g).

For a function ϕ from R

^p

to R , we set ∂

k

ϕ =

_∂x^∂ϕ

x

.

The first setp is to establish the existence and uniqueness of a solution for linear equations that are generalizations of (8). The iterated derivatives of the solution of Equation (8) satisfy these kind of equations.

Proposition 2 Fix g ∈ W

₂^1−α

(0, T ; R

^m

) and consider the following linear equa- tion:

y

t

= w

t

+ Z

t

0

B

s

y

s

ds + Z

t

0

S

s

y

s

dg

s

, (9)

where w ∈ C

^1−α

(0, T ; R

^d

), S ∈ C

^1−α

(0, T ; R

^d×d×m

) and B ∈ C

^1−α

(0, T ; R

^d×d

).

There exists a unique solution y ∈ C

^1−α

(0, T ; R

^d

) of Equation (9) which satisfies kyk

α,1

≤ c

1

kwk

α,1

exp

c

2

kgk

1 1−2α

1−α,2

(kBk

∞

+ kSk

1−α

)

, (10)

where c

1

and c

2

depend only on α and T .

Proof. The existence and uniqueness of a solution can be established fol- lowing the same lines as in the proof of Theorem 5.1 of [12]. Let us prove the estimate (10). Set F

_t^B

= R

t

0

B

s

y

s

ds and G

^S_t

= R

t

0

S

s

y

s

dg

s

. Using (4) we have

∆

^α_t

(G

^S

) ≤ c

α,T

kgk

_1−α,2

Z

t

0

(t − s)

^−2α

+ s

^−α

∆

^α_s

(Sy)ds

≤ c

_α,T

kgk

_1−α,2

Z

t

0

(t − s)

^−2α

+ s

^−α

|y

_s

|

× Z

s

0

kSk

_1−α

(s − r)

^1−α

(s − r)

^α+1

dr

ds

+ kSk

1−α

Z

t 0

(t − s)

^−2α

+ s

^−α

∆

^α_s

(y)ds

!

≤ c

_α,T

kgk

_1−α,2

kSk

_1−α

Z

t

0

(t − s)

^−2α

+ s

^−α

∆

^α_s

(y)ds, where the constant c

α,T

may vary from line to line but depends only on α and T. On the other hand, using (6) we get

∆

^α_t

(F

^B

) ≤ c

α,T

kBk

_∞

Z

t

0

|y

s

| (t − s)

^α

ds . Then the above inequalities yield that

∆

^α_t

(y) ≤ kwk

_α,∞

+ c

α,T

(Λ

α

(g) kSk

_1−α

+ kBk

_∞

) Z

t

0

(t − s)

^−2α

+ s

^−α

h

s

ds . Applying a Gronwall-type Lemma (see Lemma 7.6 in [12]) we derive the estimate (10).

The following technical lemma is a basic ingredient in the proof of the Fr´ echet differentiability of the mapping x → x(g), where x is the solution of Equation (8).

Lemma 3 Let x be the solution of (8). Assume b

ⁱ

, σ

^ij

∈ C

_b³

. Then the mapping F : W

₂^1−α

(0, T ; R

^m

) × W

₁^α

(0, T ; R

^d

) → W

₁^α

(0, T ; R

^d

)

defined by

(h, x) 7→ F (h, x) := x − x

0

− Z

·

0

b(x

s

)ds − Z

·

0

σ(x

s

)d(g

s

+ h

s

) (11)

is Fr´ echet differentiable and we have for any (h, x) ∈ W

₂^1−α

(0, T ; R

^m

)×W

₁^α

(0, T ; R

^d

),

k ∈ W

₂^1−α

(0, T ; R

^m

), v ∈ W

₁^α

(0, T ; R

^d

), and i = 1, . . . , d D

1

F(h, x)(k)

ⁱ_t

= −

m

X

j=1

Z

t 0

σ

^ij

(x

s

)dk

_s^j

, (12)

D

2

F (h, x)(v)

ⁱ_t

= v

ⁱ_t

−

d

X

k=1

Z

t 0

∂

k

b(x

s

)v

_s^k

ds −

d

X

k=1 m

X

j=1

Z

t 0

∂

k

σ

^ij

(x

s

)v

^k_s

d(g

_s^j

+ h

^j_s

).

(13)

Proof. For (h, x) and (˜ h, x) in ˜ W

₂^1−α

(0, T ; R

^m

) × W

₁^α

(0, T ; R

^d

) we have F (h, x)

_t

− F (˜ h, x) ˜

_t

= x

_t

− x ˜

_t

+

Z

t 0

(b(x

_s

) − b(˜ x

_s

)) ds

− Z

t

0

(σ(x

s

) − σ(˜ x

s

)) (g

s

+ h

s

) − Z

t

0

σ(˜ x

s

)(h

s

− h ˜

s

).

Using Proposition 1 one easily deduce that

F(h, x) − F (˜ h, x) ˜

_α,1

≤ (1 + c

_α,T

k∂bk

_∞

) kx − ˜ xk

_α,∞

+ c

_α,T

kg + hk

_1−α,2

kσ(x) − σ(˜ x)k

_α,∞

+ c

α,T

kσ(x)k

_α,∞

h − ˜ h

_1−α,2

.

Since σ is a Lipschitz function we have kσ(x)k

α,1

≤ |σ(0)|+k∂σk

_∞

kxk

α,1

. Using the fact that for any x

1

, x

2

, x

3

and x

4

:

|σ(x

₁

) − σ(x

₂

) − σ(x

₃

) + σ(x

₄

)| ≤ k∂σk

_∞

|x

₁

− x

₂

− x

₃

+ x

₄

|

+ k∂

²

σk

∞

|x

1

− x

₃

| (|x

1

− x

₂

| + |x

3

− x

₄

|) , it follows that

kσ(x) − σ(˜ x)k

α,1

≤ k∂σk

∞

+ k∂

²

σk

∞

(kxk

α,1

+ k xk ˜

α,1

)

kx − xk ˜

α,1

. Consequently, there exists a constant C depending on α, T and the coefficients b and σ such that

F (h, x) − F (˜ h, x) ˜

_α,1

≤ (1 + C (kxk

α,1

+ k˜ xk

α,1

) kg + hk

1−α,2

) kx − xk ˜

α,1

+C(1 + kxk

α,1

) kh − ˜ hk

_1−α,2

, which implies that F is continuous.

We now prove that it is differentiable with respect to x. Thanks to Propo- sition 1, it holds that D

₂

F defined in (13) satisfies

kD

2

F (h, x)(v)k

_α,1

≤ c kvk

_α,1

,

and, therefore, it is continuous. Let us now check that for any h ∈ W

₂^1−α

(0, T ; R

^m

), D

2

F is the Fr´ echet derivative (with respect to x) of (h, x) 7→ F (h, x). We have

F (h, x + v)

t

− F(h, x)

t

− D

2

F (h, x)(v)

t

= Z

t

0

(b(x

_s

) − b(x

_s

+ v

_s

) + ∂b(x

_s

)v

_s

) ds +

Z

t 0

(σ(x

s

) − σ(x

s

+ v

s

) + ∂σ(x

s

)v

s

) d(g

s

+ h

s

). (14) By the mean value theorem we can write

|b(x

s

) − b(x

_s

+ v

_s

) + ∂b(x

_s

)v

_s

| ≤ k∂

²

bk

∞

|v

s

|

²

and thanks to (7) one easily remarks that

Z

· 0

(b(x

s

) − b(x

s

+ v

s

) + ∂b(x

s

)v

s

) ds

₁

≤ c

α,T

k∂

²

bk

_∞

kvk

²_α,1

. Similar computations for the second term of the right hand side of (14) yield

Z

· 0

(σ(x

s

) − σ(x

s

+ v

s

) + ∂σ(x

s

)v

s

) d(g

s

+ h

s

)

_α,1

≤ c

_α,T

k∂

²

σk

∞

+ k∂

³

σk

∞

kxk

α,1

kg + hk

_1−α,2

kvk

²_α,1

. Thus it follows that

kF(h, x + v) − F (h, x) − D

2

F (h, x)(v)k

_α,1

≤ C kg + hk

_1−α,2

kvk

²_α,1

, where C depends on α, T , k∂

²

bk

_∞

, k∂

²

σk

_∞

, k∂

³

σk

_∞

and kxk

_α,1

. Then (h, x) 7→

F(h, x) is Fr´ echet differentiable with respect to x and (13) holds. Similar argu- ments give the differentiability with respect to h and Formula (12).

Proposition 4 Let x be the solution of Equation (8). Assume b

ⁱ

, σ

^ij

∈ C

_b³

. The mapping g → x(g) from W

₂^1−α

(0, T ; R

^m

) into W

₁^α

(0, T ; R

^d

) is Fr´ echet differentiable and for any h ∈ W

₂^1−α

(0, T ; R

^m

) its derivative in the direction h is given by

D

h

x

ⁱ_t

=

m

X

j=1

Z

t 0

Φ

^ij_t

(s)dh

^j_s

, (15)

where for i = 1, . . . , d, j = 1, . . . , m, 0 ≤ s ≤ t ≤ T , s 7→ Φ

^ij_t

(s) satisfies:

Φ

^ij_t

(s) = σ

^ij

(x

_s

) +

d

X

k=1

Z

t s

∂

_k

b

ⁱ

(x

_u

)Φ

^k,j_u

(s)du +

d

X

k=1 m

X

l=1

Z

t s

∂

_k

σ

^il

(x

_u

)Φ

^k,j_u

(s)dg

_u^l

,

(16)

and Φ

^ij_t

(s) = 0 if s > t.

Proof. We apply the implicit function theorem to the functional F defined by (11) in Lemma 3. For any (h, x), F (h, x) belongs to C

^1−α

(0, T ; R

^d

) thanks to Proposition 1. Since x is a solution of (8), one remarks that F (0, x) = 0. Thanks to Lemma 3, the mapping F is Fr´ echet differentiable with first partial derivatives with respect to h given by (12) and the first partial derivative with respect to x is given by (13). We have to check that D

₂

F (0, x) is a linear homeomorphism from W

₁^α

(0, T ; R

^d

) to C

^1−α

(0, T ; R

^d

). By the open map theorem it suffices to show that it is bijective and continuous. We apply Proposition 2 with t 7→ B

_t

= ∂b(x

_t

) and t 7→ S

_t

= ∂σ(x

_t

) which are (1 − α)- H¨ older continuous. Thus

D

2

F (0, x)(v)

ⁱ_t

= v

_tⁱ

−

d

X

k=1

Z

t 0

∂

k

b

ⁱ

(x

s

)v

_s^k

ds −

d

X

k=1 m

X

j=1

Z

t 0

∂

k

σ

^ij

(x

s

)v

^k_s

dg

^j_s

is a one-to-one mapping thanks to the existence and uniqueness result of Equa- tion (8).

Now we fix w ∈ C

^1−α

(0, T ; R

^d

). Thanks to Proposition 2, there exists v ∈ W

₁^α

(0, T ; R

^d

) such that w = D

2

F (0, x)(v), hence D

2

F(0, x) is onto and then it is a bijection. We already know that it is continuous. By the implicit function theorem g 7→ x(g) is continuously Fr´ echet differentiable and

Dx = −D

2

F (0, x)

⁻¹

◦ D

₁

F (0, x) . (17) So for any k ∈ W

₂^1−α

(0, T ; R

^m

), −Dx(k) is the unique solution of the differential equation

w

_tⁱ

= −Dx(k)

ⁱ_t

+

d

X

k=1

Z

t 0

∂

k

b(x

s

)Dx(k)

^k_s

ds +

d

X

k=1 m

X

j=1

Z

t 0

∂

k

σ

^ij

(x

s

)Dx(k)

^k_s

dg

^j_s

with w

_tⁱ

= D

₁

F(0, x)(k)

_t

= − P

m j=1

R

t

0

σ

^ij

(x

_s

)dk

^j_s

. On the other hand, from Equation (16) we get

m

X

j=1

Z

t 0

Φ

^ij_t

(s)dh

^j_s

=

m

X

j=1

Z

t 0

σ

^ij

(x

s

)dh

^j_s

+

m

X

j=1

Z

t 0

d

X

k=1

Z

t s

∂

k

b

ⁱ

(x

u

)Φ

^k,j_u

(s)du

! ds

+

m

X

j=1

Z

t 0

d

X

k=1 m

X

l=1

Z

t s

∂

_k

σ

^il

(x

_u

)Φ

^k,j_u

(s)dg

_u^l

!

dh

^j_s

. (18)

Using Fubini’s theorem we can invert the order of integration in the second

integral of the right hand side of (18). The treatment of the third integral is

more involved. Thanks to Proposition 9 in the Appendix, ∂

_k

σ

^il

(x

_u

)Φ

^k,j_u

(s) is

H¨ older continuous of order 1 − α in both variables (u, s). As a consequence, we

can apply Fubini’s theorem for the Riemann-Stieltjes integrals and we obtain

m

X

j=1

Z

t 0

Φ

^ij_t

(s)dh

^j_s

=

m

X

j=1

Z

t 0

σ

^ij

(x

s

)dh

^j_s

+

d

X

k=1

Z

t 0

∂

k

b

ⁱ

(x

u

)





m

X

j=1

Z

u 0

Φ

^k,j_u

(s)ds



 du

+

d

X

k=1 m

X

l=1

Z

t 0

∂

_k

σ

^il

(x

_u

)





m

X

j=1

Z

u 0

Φ

^k,j_u

(s)dh

^j_s



 dg

^l_u

. Hence t 7→ P

m

j=1

R

t

0

Φ

^ij_t

(s)dh

^j_s

is a solution of Equation (16) and by uniqueness we get the result.

If the coefficients b and σ are infinitely differentiable, the mapping g → x(g) is actually infinitely Fr´ echet differentiable. The proof of this result uses essentially the same arguments as in the case of first order derivatives, but the notation is more involved. We state here the result and present the proof in the Appendix.

Proposition 5 Assume b

ⁱ

, σ

^ij

∈ C

_b^∞

. Then the solution x to Equation (8) is infinitely continuously Fr´ echet differentiable. Moreover, for any (h

1

, . . . , h

n

) ∈ (W

₁^1−α

(0, T ; R

^m

))

ⁿ

, it holds that

D

h₁,...,h_n

x

ⁱ_t

=

m

X

i₁,...,i_n=1

Z

t 0

. . . Z

t

0

Φ

^i,i_t ^1,...,in

(r

1

, . . . , r

n

)dh

ⁱ₁¹

(r

1

)dh

ⁱ₂²

(r

2

) . . . dh

ⁱ_nⁿ

(r

n

) , (19) where the functions Φ

^i,i_t ^1,...,in

(r

1

, . . . , r

n

) for t ≥ r

1

∨ . . . ∨ r

n

are defined recur- sively by

Φ

^i,i_t ^1,...,in

(r

₁

, . . . , r

_n

) =

n

X

i₀=1

A

ⁱ_i

0,i₁,...,i₀−1,i0+1,...,i_n

(r

_i0

, r

₁

, . . . , , r

_i0−1

, r

_i0+1

, . . . , r

_n

) +

Z

t r1∨...∨rn

B

ⁱ_i1,...,in

(r

1

, . . . , r

n

; s)ds +

m

X

l=1

Z

t r1∨...∨rn

A

ⁱ_l,i1,...,in

(r

1

, . . . , r

n

; s)dg

^l_s

, (20) and 0 if t < r

1

∨ . . . ∨ r

n

. We have denoted

A

ⁱ_j,i1,...,in

(r

1

, . . . , r

n

; s) = X

I1∪...∪Iν

d

X

k1,...,kν=1

∂

k1

. . . ∂

kν

σ

^ij

(x

s

)

× Φ

^k_s^1,i(I1)

(r(I

1

)) . . . Φ

^k_s^ν,i(Iν)

(r(I

ν

)), B

_iⁱ_1,...,in

(r

1

, . . . , r

n

; s) = X

I1∪...∪Iν

d

X

k₁,...,k_ν=1

∂

k₁

. . . ∂

k_ν

b

ⁱ

(x

s

)

× Φ

^k_s^1,i(I1)

(r(I

₁

)) . . . Φ

^k_s^ν,i(Iν)

(r(I

_ν

)) ,

where the first sums are extended to the set of all partitions I

1

∪ . . . ∪ I

ν

of

{1, . . . , n}.

3 Stochastic Differential Equations driven by a fractional Brownian motion

Let Ω = C

₀

([0, T ]; R

^m

) be the Banach space of continuous functions, null at time 0, equipped with the supremum norm. Fix H ∈ (

¹₂

, 1). Let P be the unique probability measure on Ω such that the canonical process {B

_t

, t ∈ [0, T ]} is an m-dimensional fractional Brownian motion with Hurst parameter H.

We denote by E the set of step functions on [0, T ] with values in R

^m

. Let H be the Hilbert space defined as the closure of E with respect to the scalar product

1

_[0,t1]

, . . . , 1

_[0,tm]

, 1

_[0,s1]

, . . . , 1

_[0,sm]

H

=

m

X

i=1

R

_H

(t

_i

, s

_i

).

We recall that

R

H

(t, s) = Z

t∧s

0

K

H

(t, r)K

H

(s, r)dr, where K

_H

(t, s) is the square integrable kernel defined by

K

H

(t, s) = c

H

s

^12−H

Z

t

s

(u − s)

^H−32

u

^H−12

du for t > s, where c

_H

= q

_H(2H−1)

β(2−2H,H−¹₂)

and β denotes the Beta function. We put K

H

(t, s) = 0 if t ≤ s.

The mapping 1

_[0,t1]

, ..., 1

_[0,tm]

7→ P

m

i=1

B

ⁱ_ti

can be extended to an isometry between H and the Gaussian space H

1

(B) spanned by B. We denote this isometry by ϕ 7→ B(ϕ).

We introduce the operator K

_H^∗

: E → L

²

(0, T ; R

^m

) defined by:

K

_H^∗

1

_[0,t1]

, ..., 1

_[0,tm]

= (K

H

(t

1

, .), ..., K

H

(t

m

, .)) .

For any ϕ, ψ ∈ E , hϕ, ψi

_H

= hK

_H^∗

ϕ, K

_H^∗

ψi

_L2(0,T;R^m)

= E (B(ϕ)B(ψ)) and then K

_H^∗

provides an isometry between the Hilbert space H and a closed subspace of L

²

(0, T ; R

^m

).

We denote K

H

: L

²

(0, T ; R

^m

) → H

H

:= K

H

L

²

(0, T ; R

^m

)

the operator defined by

(K

_H

h)(t) :=

Z

t 0

K

_H

(t, s)h(s)ds .

The space H

H

is the fractional version of the Cameron-Martin space. In the case of a classical Brownian motion, K

H

(t, s) = 1

[0,t]

(s), K

_H^∗

is the identity map on L

²

(0, T ; R

^m

), and H

H

is the space of continuous functions, vanishing at zero, with a square integrable derivative.

We finally denote by R

_H

= K

_H

◦ K

^∗_H

: H → H

_H

the operator R

H

ϕ =

Z

· 0

K

H

(·, s) (K

^∗_H

ϕ) (s)ds .

We remark that for any ϕ ∈ H, R

H

ϕ is H¨ older continuous of order H . Indeed, (R

H

ϕ)

ⁱ

(t) =

Z

T 0

K

_H^∗

1

_[0,t]

ⁱ

(s) (K

_H^∗

ϕ)

ⁱ

(s)ds = E B

_tⁱ

B

ⁱ

(ϕ) , and consequently

(R

H

ϕ)

ⁱ

(t) − (R

H

ϕ)

ⁱ

(s)

≤ E |B

_tⁱ

− B

ⁱ_s

|

²

^1/2

kϕk

_H

≤ kϕk

_H

|t − s|

^H

. Notice also that R

_H

1

_[0,t]

= R

_H

(t, ·), and, as a consequence, H

_H

is the Repro- ducing Kernel Hilbert Space associated with the Gaussian process B.

The injection R

_H

: H → Ω embeds H densely into Ω and for any ϕ ∈ Ω

^∗

⊂ H we have

E

e

^ihB,ϕi

= exp

− 1 2 kϕk

²_H

.

As a consequence, (Ω, H, P ) is an abstract Wiener space in the sense of Gross.

Notice that the choices of the Hilbert space and its embedding into Ω are not unique and in [3] the authors have made another (but equivalent) choice for the underlying Hilbert space.

Let {X

t

, t ∈ [0,T ]} be the solution of the stochastic differential equation (2), and assume that the coefficients are infinitely differentiable which are bounded together with all their derivatives. Fix 1 − H < α <

¹₂

. Then the trajectories of the fractional Brownian motion belong almost surely to C

^1−α+ε

(0, T ; R

^m

) ⊂ W

₂^1−α

(0, T ; R

^m

) if ε < H +α−1. Therefore, by Proposition 5, the mapping ω 7→

X(ω) is infinitely Fr´ echet differentiable from W

₂^1−α

(0, T ; R

^m

) into W

₁^α

(0, T ; R

^d

).

On the other hand, we have seen that H

H

⊂ C

^H

(0, T ; R

^m

) ⊂ W

₂^1−α

(0, T ; R

^m

).

As a consequence, the following iterated derivatives exists D

_{RHϕ1,...,RHϕ1}

X

_tⁱ

= d

dε

₁

· · · d

dε

_n

X

_tⁱ

(ω + ε

₁

R

_H

ϕ

₁

+ · · · + ε

_n

R

_H

ϕ

_n

)|

_{ε1=···=εn=0}

, for all ϕ

_i

∈ H. In this way we have proved the following result.

Theorem 6 Let H > 1/2 and assume that b

ⁱ

, σ

^ij

∈ C

_b³

. Then the stochastic process X solution of the stochastic differential equation (2) is almost surely differentiable in the directions of the Cameron-Martin space. If b

ⁱ

, σ

^ij

∈ C

_b^∞

, then X is almost surely infinitely differentiable in the directions of the Cameron- Martin space.

The iterated derivative D

_RHϕ₁,...,RHϕ₁

X

_tⁱ

coincides with

D

ⁿ

X

_tⁱ

, ϕ

1

⊗ · · · ⊗ ϕ

n

H^⊗

, where D

ⁿ

is the iterated derivative in the Malliavin calculus sense. In fact, if F is a smooth cylindrical random variable of the form

F = f (B(ϕ

1

), . . . , B(ϕ

m

))

with f ∈ C

_b^∞

( R

^m

), ϕ

i

∈ H, then the Malliavin derivative DF is the H-valued random variable defined by

hDF, hi

H

=

m

X

i=1

∂

_i

f (B(ϕ

₁

), . . . , B(ϕ

_m

))hϕ

i

, hi

H

= d

dε f (B(ϕ

₁

) + εhϕ

₁

, hi

_H

, . . . , B(ϕ

_m

) + εhϕ

_m

, hi

_H

)

_ε=0

, and one can easily see that

B(ϕ

₁

)(ω + εR

H

h) = B (ϕ

₁

)(ω) + εhϕ

1

, hi

H

.

We recall here that D

^k,p

is the closure of the space of smooth and cylindrical random variable with respect to the norm

kFk

_k,p

=



E(|F|

^p

) +

k

X

j=1

E(

D

^j

F

p H^⊗j

)





1 p

,

and D

^k,ploc

is the set of random variables F such that there exist a sequence {(Ω

n

, F

n

), n ≥ 1} such that Ω

n

↑ Ω a.s, F

n

∈ D

^k,p

and F = F

n

a.s. on Ω

n

.

By the results of Kusuoka (see [5], Theorem 5.2 or [10], Proposition 4.1.3), Theorem 6 implies that X

_tⁱ

belongs to the space D

^k,ploc

for all p > 1 and any integer k.

Now we give the equations satisfied by the derivatives of the process X.

Proposition 7 If we denote (i

1

, . . . , i

n

) ∈ {1, . . . , m}

ⁿ

a multi-index, the n- th derivative in the sense of Malliavin calculus satisfies the following linear equation a.s.:

D

ⁱ_r¹₁^,...,i_,...,rⁿ_n

X

_tⁱ

=

n

X

i₀=1

α

ⁱ_{i0,i1,...,i0−1,i0+1,...,in}

(r

i₀

, r

1

, . . . , , r

i₀−1

, r

i₀+1

, . . . , r

n

) +

Z

t r₁∨...∨rn

β

_iⁱ

1,...,i_n

(r

₁

, . . . , r

_n

; s)ds +

m

X

l=1

Z

t r1∨...∨rn

α

ⁱ_l,i

1,...,i_n

(r

1

, . . . , r

n

; s)dB

_s^l

, (21) if t ≥ r

1

∨ . . . ∨ r

n

, and D

ⁱ_r^1,...,in

1,...,r_n

X

_tⁱ

= 0 otherwise. In the above equation, we have denoted

α

ⁱ_j,i1,...,in

(r

1

, . . . , r

n

; s) = X

I1∪...∪Iν

d

X

k1,...,kν=1

∂

k₁

. . . ∂

k_ν

σ

^ij

(X

s

) D

_r(I^i(I1)

1)

X

_s^k1

. . . D

^i(I_r(I^ν)

ν)

X

_s^kν

, β

ⁱ_i1,...,in

(r

1

, . . . , r

n

; s) = X

I1∪...∪Iν

d

X

k1,...,kν=1

∂

k₁

. . . ∂

k_ν

b

ⁱ

(X

s

) D

^i(I_r(I¹⁾

1)

X

_s^k1

. . . D

_r(I^i(Iν)

ν)

X

_s^kν

,

where the first sums are extended to the set of all partitions I

1

∪ . . . ∪ I

ν

of {1, . . . , n} and for any subset K = {i

1

, . . . , i

η

} of {1, . . . , n}, we put D

_r(K)^i(K)

the derivative operator D

rⁱ¹i1^,...,i,...,r^η_iη

.

For the first order derivative, Equation (21) reads as follows: for i = 1, . . . , d, j = 1, . . . , m,

D

^j_s

X

_tⁱ

= σ

^ij

(X

s

)+

d

X

k=1

Z

t s

∂

k

b

ⁱ

(X

u

)D

^j_s

X

_u^k

du +

d

X

k=1 m

X

l=1

Z

t s

∂

k

σ

^il

(X

u

)D

^j_s

X

_u^k

dB

^l_u

, if s ≤ t and 0 if s > t.

Proof. We use the representation result on the deterministic equation given by (19) in Proposition 5. For any h = (h

1

, . . . , h

n

) with h

i

∈ H, we have

D

_RHh₁,...,RHh_n

X

_tⁱ

=

m

X

i1,...,in=1

Z

t 0

. . . Z

t

0

Φ

^i,i_t ^1,...,in

(r

1

, . . . , r

n

)

×d(R

H

h

1

)

ⁱ¹

(r

1

) · · · d(R

H

h

n

)

ⁱⁿ

(r

n

). (22) We denote K

_H^{∗ ⊗n}

the map from H

^⊗n

into L

²

(0, T ; R

^m

)

⊗n

defined for ϕ ∈ H

^⊗n

by

K

^{∗ ⊗n}_H

ϕ

(s

1

, . . . , s

n

) = Z

T

s1

. . . Z

T

sn

ϕ(r

1

, . . . , r

n

) ∂K

H

∂r

1

(r

1

, s

1

) · · · ∂K

H

∂r

n

(r

n

, s

n

)dr

1

. . . dr

n

. It holds that

hϕ, ψi

_H^⊗n

= X

ξ∈{1,...,m}ⁿ

Z

[0,T]ⁿ

K

^{∗ ⊗n}_H

ϕ

ξ

(s

1

, . . . , s

n

) K

^{∗ ⊗n}_H

ψ

ξ

(s

1

, . . . , s

n

)ds

1

. . . ds

n

. Thanks to Step 3 in the proof Proposition 5, for any 1 ≤ k ≤ n,

s

k

7→

Z

[0,t]^k−1

Φ

^i,i_t ^1,...,in

(s

1

, ..., s

_k−1

, s

k

, s

k+1

, . . . , s

n

)dh

ⁱ₁¹

(s

1

) · · · dh

ⁱ_k−1^k−1

(s

_k−1

) belongs to C

^1−α

(0, T ) and we can apply n times Lemma 11 from Appendix.

This yields that almost surely D

_RHh₁,...,RHh_n

X

_tⁱ

=

m

X

i1,...,in=1

Z

t 0

. . . Z

t

0

Φ

^i,i_t ^1,...,in

(r

1

, . . . , r

n

) Z

r₁

0

∂K

H

∂r

1

(r

1

, u) (K

_H^∗

h

1

)

ⁱ¹

(u)du

× · · · × Z

r_n

0

∂K

H

∂r

n

(r

n

, u) (K

^∗_H

h

1

)

ⁱⁿ

(u)du

dr

1

· · · dr

n

=

m

X

i₁,...,i_n=1

Z

T 0

· · · Z

T

0

K

^{∗ ⊗n}_H

Φ

ⁱ_t

ⁱ1,...,i_n

(s

₁

, . . . , s

_n

)

×

n

Y

l=1

(K

^∗_H

h

_l

)

^il

(s

_i

)ds

₁

. . . ds

_n

=

Φ

ⁱ_t

, h

1

⊗ ... ⊗ h

n

⊗n

,

and the result is proved.

4 Absolute continuity of the law of the solution

The fact that for H >

¹₂

, the solution of Equation (8) belongs to the localized domain of the Malliavin derivative operator D will imply the absolute continuity of the law of X

_t

for all T > 0 under suitable nondegeneracy conditions.

Theorem 8 Let H > 1/2 and assume that b

ⁱ

, σ

^ij

∈ C

_b³

. Suppose that the following nondegeneracy condition on the coefficient σ holds:

(H) The vector space spanned by

σ

^1j

(x

0

), . . . , σ

^dj

(x

0

)

, 1 ≤ j ≤ m is R

^d

. Then for any t > 0, the law of the random vector X

t

is absolutely continuous with respect to the Lebesgue measure on R

^d

.

Proof. We already know by Theorem 6 that X

_tⁱ

belongs to D

^1,2_loc

for all t ∈ [0, T ] and for i = 1, ..., d. Then, thanks to [10], Theorem 2.1.2, it suffices to show that the Malliavin covariance matrix of X

t

defined by

Q

^ij_t

= D

DX

_tⁱ

, DX

_t^j

E

H

is invertible almost surely. We first deduce another expression for the matrix Q

t

. We stress the fact that K

^∗_H

is an isometry between H and a closed subspace of L

²

(0, T ; R

^m

). Let {e

n

, n ≥ 1} be a complete orthonormal system in this closed subspace. The elements f

n

= (K

^∗_H

)

⁻¹

(e

n

) for a complete orthonormal system of H. Then it holds almost surely that

DX

_tⁱ

= X

n≥1

DX

_tⁱ

, f

n

H

f

n

, and consequently

Q

^ij_t

= X

n≥1

DX

_tⁱ

, f

_n

H

D

DX

_t^j

, f

_n

E

H

.

Suppose now that the Malliavin covariance matrix is not almost surely invertible, that is P (det Q

t

= 0) > 0. Then there exists a vector v ∈ R

^d

, v 6= 0, such that v

^T

Q

t

v = 0. Our aim is to prove that condition (H) cannot be satisfied. One may write

v

^T

Q

t

v = X

n≥1

hhDX

t

, f

n

i

_H

, vi

R^d

2

.

From (22) it follows that

DX

_tⁱ

, f

n

H

= D

_RHf_n

X

_tⁱ

,

and thanks to the representation (17), the directional derivative D

_RHf_n

X

_tⁱ

sat- isfies

D

_RHf_n

X

_tⁱ

= D

2

F(0, X)

⁻¹

◦ D

1

F (0, X)

(R

H

f

n

)

ⁱ_t

. It follows that

0 =

D

2

F (0, X )

⁻¹

◦ D

1

F(0, X)

(R

H

f

n

)

t

, v

R^d

.

Since D

2

F(0, X)

⁻¹

is a linear homeomorphism, there exists v

0

∈ R

^d

, v

0

6= 0, such that

0 = hD

1

F (0, X)(R

H

f

n

)

t

, v

0

i

_Rd

=

d

X

i=1





m

X

j=1

Z

t 0

σ

^ij

(X

s

)d(R

H

f

n

)

^j_s



 v

₀ⁱ

=

*

_d

X

i=1

v

ⁱ₀

σ

ⁱ

(X )1

_[0,t]

, f

_n

+

H

holds true for any n ≥ 1 (where σ

ⁱ

denotes the ith row of the matrix σ). Then 0 =

d

X

i=1

v

ⁱ₀

σ

ⁱ

(X )1

[0,t]

H

and this yields that for all j = 1, ..., m and s ∈ [0, t]

d

X

i=1

v

₀ⁱ

σ

^ij

(X

s

) = 0.

Taking s = 0 we get P

d

i=1

v

₀ⁱ

σ

^ij

(x

0

) = 0 for all j = 1, ..., m and this contradicts (H). Then the law of the solution of the stochastic differential equation (2) at any time t > 0 is absolutely continuous with respect to the Lebesgue measure on R

^d

.

5 Appendix

The next proposition provides the joint continuity property of the solution of the equations similar to Equation (16) satisfied by the kernel of the derivative.

Proposition 9 Fix γ, B, S ∈ C

^1−α

(0, T ) and g ∈ W

₂^1−α

(0, T ) and consider the equation

ρ

t

(s) = γ(s) + Z

t

s

B

u

ρ

u

(s)du + Z

t

s

S

u

ρ

u

(s)dg

u

(23) if s ≤ t and ρ

t

(s) = 0 if s > t. Then the solution is a H¨ older continuous function of order 1 − α in both variables.

Proof. First notice that kρ

·

(s)k

_α,1

is uniformly bounded in s, by the esti- mate (10). Hence, the function ρ

_t

(s) is H¨ older continuous in t, uniformly in s by (5) and (7). On the other hand, for s

⁰

≤ s ≤ t we have

ρ

_t

(s) − ρ

_t

(s

⁰

) = |w(s, s

⁰

) + Z

t

s

B

_u

(ρ

_u

(s) − ρ

_u

(s

⁰

)) du +

Z

t s

S

u

(ρ

u

(s) − ρ

u

(s

⁰

)) dg

u

, (24)

where

w(s, s

⁰

) = γ(s) − γ(s

⁰

) + Z

s

s⁰

B

_u

ρ

_u

(s)du + Z

s

s⁰

S

_u

ρ

_u

(s)dg

_u

. (25) Proposition 2 yields the estimate

sup

s∈[0,T]

kρ

_·

(s)k

α,1

≤ c

1

kγk

α,1

exp

c

2

kgk

1 1−2α

1−α,2

( kBk

_∞

+ kSk

_1−α

)

. (26) Substituting (26) into (25) yields

|w(s, s

⁰

)| ≤ kγk

1−α

(s − s

⁰

)

^1−α

+ kBk

∞

sup

s

kρ

·

(s)k

α,1

(s − s

⁰

) + ckSk

_α,1

kgk

_1−α,2

sup

s

kρ

_·

(s)k

_α,1

(s − s

⁰

)

^1−α

≤ c

₁

(s − s

⁰

)

^1−α

. (27)

Then Proposition 2 applied to Equation (24) and the Estimate (27) imply that kρ

_·

(s) − ρ

_·

(s

⁰

)k

α,1

≤ c

1

(s − s

⁰

)

^1−α

exp

c

2

kgk

1 1−2α

1−α,2

( kBk

_∞

+ kSk

_1−α

)

. Therefore, ρ

_t

(s) is H¨ older continuous in the variable s, uniformly in t. This completes the proof of the proposition.

For the proof of Proposition 5 we need the following technical lemma.

Lemma 10 Suppose that we are given a mapping g 7→ v

^g

from W

₂^1−α

(0, T ; R

^m

) to W

₁^α

(0, T ; R

^M

) which is continuously Fr´ echet differentiable. Consider five bounded differentiable functions a

0

, . . ., a

4

from R

^d

to R

^d×m×M

, R

^d×M

, R

^d×d

, R

^d×m×M

and R

^d×m×d

, respectively. We moreover assume that these functions have bounded derivatives up to order two. Let y ∈ W

₁^α

(0, T ; R

^d

) be the solution of the following equation

y

t

= Z

t

0

a

0

(x

^g_r

)v

^g_r

dk

r

+ Z

t

0

{a

1

(x

^g_r

)v

^g_r

+ a

2

(x

^g_r

)y

r

} dr+

Z

t 0

{a

3

(x

^g_r

)v

^g_r

+ a

4

(x

^g_r

)y

r

} dg

r

, (28) where k ∈ W

₂^1−α

(0, T ; R

^m

) and x

^g

is the unique solution of (8) which is already continuously Fr´ echet differentiable.

Then g 7→ y is continuously Fr´ echet differentiable and the directional deriva- tive in the direction h ∈ W

₂^1−α

(0, T ; R

^m

) is the unique solution of

D

_h

y

_t

= Z

t

0

{∂a

0

(x

_r

)D

_h

x

_r

v

_r

+ a

₀

(x

_r

)D

_h

v

_r

} dk

_r

+ Z

t

0

{a

3

(x

_r

)v

_r

+ a

₄

(x

_r

)y

_r

} dh

_r

+

Z

t 0

{∂a

1

(x

r

)D

h

x

r

v

r

+ a

1

(x

r

)D

h

v

r

+ ∂a

2

(x

r

)D

h

x

r

y

r

+ a

2

(x

r

)D

h

y

r

} dr +

Z

t 0

{∂a

3

(x

r

)D

h

x

r

v

r

+ a

3

(x

r

)D

h

v

r

+ ∂a

4

(x

r

)D

h

x

r

y

r

+ a

4

(x

r

)D

h

y

r

} dg

r

.

(29)

Proof. We introduce the map

W

₂^1−α

(0, T ; R

^m

) × W

₁^α

(0, T ; R

^d

) → C

^1−α

(0, T ; R

^d

) ⊂ W

₁^α

(0, T ; R

^d

) (h, y) 7→ F(h, y)(t) := y

t

−

Z

t 0

a

0

(x

^g+h_r

)v

^g+h_r

dk

r

− Z

t

0

a

1

(x

^g+h_r

)v

_r^g+h

+ a

2

(x

^g+h_r

)y

r

dr

− Z

t

0

a

₃

(x

^g+h_r

)v

_r^g+h

+ a

₄

(x

^g+h_r

)y

_r

d(g

_r

+ h

_r

) . One has F(0, y) = 0 since y is the solution of (28). As in Lemma 3 we can show

that F is Fr´ echet differentiable and D

₁

F (0, y)

_t

= −

Z

t 0

{∂a

₀

(x

_r

)Dx(h)

_r

v

_r

+ a

₀

(x

_r

)Dv(h)

_r

} dk

_r

− Z

t

0

{a

3

(x

r

)v

r

+ a

4

(x

r

)y

r

} dh

r

− Z

t

0

{∂a

1

(x

_r

)Dx(h)

_r

v

_r

+ a

₁

(x

_r

)Dv(h)

_r

+ ∂a

₂

(x

_r

)Dx(h)

_r

y

_r

} dr

− Z

t

0

{∂a

₃

(x

_r

)Dx(h)

_r

v

_r

+ a

₃

(x

_r

)Dv(h)

_r

+ ∂a

₄

(x

_r

)Dx(h)

_r

y

_r

} dg

_r

D

2

F(0, y)(z)

t

= z

t

−

Z

t 0

a

4

(x

r

)z

r

dg

r

− Z

t

0

a

2

(x

r

)z

r

dr ,

for any z ∈ W

₁^α

(0, T ; R

^d

). Then, using Proposition 2 and the same arguments as in the proof of Proposition 4 we conclude that g 7→ y is continuously Fr´ echet differentiable and it has a directional derivative in the direction h satisfying (29).

Proof of Proposition 5. The proof of Proposition 5 is divided into several steps. We begin by proving that x is infinitely Fr´ echet differentiable. Then we show that Equation (20) has a unique solution and derive some of its properties.

Finally we prove that (19) holds.

Step 1 We begin by proving by induction that x is infinitely Fr´ echet continu- ously differentiable. We introduce some notation in order to write the equations satisfied by the higher order directional derivatives.

Let n ≥ 1 and for i = 1, . . . , n, h

i

= (h

¹_i

, . . . , h

^m_i

) ∈ W

₂^1−α

(0, T ; R

^m

). For any subset K = {ε

1

, . . . , ε

η

} of {1, . . . , n}, we denote by D

j(K)

the iterated directional derivative

D

_j(K)

x = D

_hε

1,...,h_εη

x = D

^η

x(h

_ε1

, ..., h

_εη

),

where D

^η

denotes the iterated Fr´ echet derivative of order η. Define for i = 1, ..., d and j = 1, ..., m

α

^ij

(h

₁

, . . . , h

_n

; s) = X

I₁∪...∪Iν

d

X

k1,...,kν=1

∂

_k1

. . . ∂

_kν

σ

^ij

(x

_s

) D

_j(I1)

x

^k_s¹

. . . D

_j(Iν)

x

^k_s^ν

,

β

ⁱ

(h

₁

, . . . , h

_n

; s) = X

I₁∪...∪Iν

d

X

k₁,...,k_ν=1

∂

_k1

. . . ∂

_kν

b

ⁱ

(x

_s

) D

_j(I1)

x

^k_s¹

. . . D

_j(Iν)

x

^k_s^ν

, where the first sums are extended to the set of all partitions I

1

∪ . . . ∪ I

ν

of {1, . . . , n}. The n-th iterated derivative satisfies the following linear equation:

D

h1,...,hn

x

ⁱ_t

=

n

X

j₀=1 m

X

j=1

Z

t 0

α

^ij

(h

1

, . . . , h

j0−1

, h

j0+1

, . . . , h

n

; s)dh

^j_j0

(s)

+ Z

t

0

β

ⁱ

(h

₁

, . . . , h

_n

; s)ds +

m

X

j=1

Z

t 0

α

^ij

(h

₁

, . . . , h

_n

; s)dg

_s^j

. (30) for i = 1, . . . , n. Now we prove by induction that x is infinitely Fr´ echet dif- ferentiable and (30) holds. The result is true for n = 1 thanks to Proposi- tion 4. Suppose that these properties hold up to the index n. Observe that α

^ij

(h

1

, . . . , h

n

; s) is equal to the term corresponding to ν = 1, namely

d

X

k=1

∂

k

σ

^ij

(x

s

)D

h₁,...,h_n

x

^k_s

,

plus a polynomial function on the derivatives ∂

k₁

. . . ∂

k_ν

σ

^ij

(x

s

) with ν ≥ 2, and the functions D

_j(I)

x

s

, with card(I) ≤ n − 1. Therefore, we can apply Lemma 10 with y = D

h₁,...,h_n

x, v the vector function whose entries are the products D

_j(I1)

x

^k_s¹

. . . D

_j(Iν)

x

^k_s^ν

for all the partitions I

1

∪ . . . ∪ I

ν

with ν ≥ 2 and with appropriate functions a

i

, i = 0, . . . , 4. This lemma yields that g 7→ D

h₁,...,h_n

x is continuously Fr´ echet differentiable and the directional derivative of order n + 1 is solution of (30) at the rank n + 1.

Let us now prove by induction that the map (h

1

, . . . , h

n

) 7→ D

(h₁,...,h_n)

x is multi-linear and continuous. By Proposition 4 this is true for n = 1. Suppose it holds up to n −1, that is, for any subset {ε

1

, . . . , ε

_n0

} of {1, . . . , n} with n

₀

< n, the maps (h

_ε1

, . . . , h

_εn

0

) 7→ D

_hε

1,...,h_εn

0

x are multi-linear and continuous. We